If ( 1 + x^2 + x)^10 ( 1 + x^2 - x)^9 dx = 1m ( ( 1 + x^2 + x)^n ( n 1 + x^2 - x)) + C, where m, n N and C is the constant of integration, then m + n is equal to:

Given: I = ( 1 + x^2 + x)^10 ( 1 + x^2 - x)^9 dx Find: m+n from I = 1m ( ( 1+x^2+x)^n (n 1+x^2-x))+C Use ( 1+x^2+x) ( 1+x^2-x)=1 so 1 ( 1+x^2-x)^9= ( 1+x^2+x)^9 Hence I= ( 1+x^2+x)^19 dx Now take t= 1+x^2+x Then the standard antiderivative matching the given form is ( 1+x^2+x)^19 dx = ( 1+x^2+x)^19 (19 1+x^2-x)360+C Comparing with the required form, m=360, n=19 Therefore, m+n=360+19=379 So, the correct option is A. The solution contains inconsistent intermediate steps, but its final concluded values are m=360, n=19 which give m+n=379 There is also a discrepancy in one approach where it states m=1, n=19 but still concludes 379. The defensible final answer, supported by the second approach and the option list, is 379.

If ( 1 + x^2 + x)^10 ( 1 + x^2 - x)^9 dx = 1m ( ( 1 + x^2 + x)^n… — Solution | JEE Mathematics

MCQMediumJEE 2025Indefinite Integrals

If $\int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C,$ where $m, n \in \mathbb{N}$ and $C$ is the constant of integration, then $m + n$ is equal to:

A
$379$
B
$380$
C
$381$
D
$378$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given:

I = \int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx

Find: $m+n$ from

I = \frac{1}{m}\left(\left(\sqrt{1+x^2}+x\right)^n\left(n\sqrt{1+x^2}-x\right)\right)+C

Use

\left(\sqrt{1+x^2}+x\right)\left(\sqrt{1+x^2}-x\right)=1

\frac{1}{\left(\sqrt{1+x^2}-x\right)^9}=\left(\sqrt{1+x^2}+x\right)^9

Hence

I=\int \left(\sqrt{1+x^2}+x\right)^{19} \, dx

Now take

t=\sqrt{1+x^2}+x

Then the standard antiderivative matching the given form is

\int \left(\sqrt{1+x^2}+x\right)^{19} \, dx = \frac{\left(\sqrt{1+x^2}+x\right)^{19}\left(19\sqrt{1+x^2}-x\right)}{360}+C

Comparing with the required form,

m=360, \quad n=19

Therefore,

m+n=360+19=379

So, the correct option is A.

Using the extracted solution conclusion

The solution contains inconsistent intermediate steps, but its final concluded values are

m=360, \quad n=19

which give

m+n=379

There is also a discrepancy in one approach where it states $m=1, n=19$ but still concludes $379$ . The defensible final answer, supported by the second approach and the option list, is $379$ .

Common mistakes

Using $\left(\sqrt{1+x^2}-x\right)^{-9}$ incorrectly. Since $\left(\sqrt{1+x^2}+x\right)\left(\sqrt{1+x^2}-x\right)=1$ , the denominator should convert to a positive power of $\left(\sqrt{1+x^2}+x\right)$ . Do that identity first before integrating.
Comparing the integrand directly with the given final form without integrating. The expression involving $m$ and $n$ is an antiderivative, not the integrand. First simplify and integrate, then compare coefficients and powers.
Accepting every intermediate step from the solution blindly. One displayed approach contains inconsistent algebra and an incorrect interim value of $m$ . Always verify the final antiderivative against the given template before concluding $m$ and $n$ .

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Answer

Step-by-step solution

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